Figure 1

Off-lattice rhombi tile the plane: Constant-pressure simulations of hard rhombi of aspect ratio $A=\sqrt{3}$ show the emergence of long-range hexatic order at densities $\rho$ above about 70%. Here and subsequently, rhombi are colored according to their absolute orientations. Left: Model geometry, overlaid by the isosurface of TPTC ground-state electron density.Reuse & Permissions

Figure 2

Rhombus tiles with attractive H-bond interactions self-assemble into random and ordered solid phases. Regular rhombi ($A=\sqrt{3}$) assemble as a random solid; sufficiently irregular rhombi form the parallel ordered one (see top right; error bars denote 1 standard deviation). Snapshot enlargements show that the ordered phase emerges when the 3-particle boxes required to form the random tiling become geometrically strained.Reuse & Permissions

Figure 3

Design space of molecular rhombi. Schematic phase diagram of aspect ratio $A$ and patch placement $\Delta$ (phase boundaries are approximate). DFT calculations reveal that TPTC (right) corresponds to a nearly regular rhombus whose patches are placed near-centrally.Reuse & Permissions

Figure 4

Interactions other than H-bond attractions may act to select tiling phases. (a) We consider regular rhombi interacting nonspecifically, initially arranged as a random tiling of a triangular lattice. We allow sampling of dense phases using the 3-particle rotation algorithm (see Ref. 14); we plot the thermal average of the tiling order parameter $\Psi$ as a function of ${ϵ}_n$, the strength of the nonspecific attraction (the peaked line is shifted and scaled variance of $\Psi$). This attraction favors high-order vertices of the kind found readily in the nonparallel ordered tiling (inset bottom left: Particle $i$ has two lots of five vertex neighbors), and if strong enough it destabilizes the random tiling. Inset top right: Average number $n_n$ of pairwise nonspecific interactions made by particles. (b) Self-assembly of regular and irregular rhombi driven by the nonspecific interaction. We show configurations taken from dynamical trajectories of rhombi equipped with nonspecific attractions of fixed strength, for the cases $A^2=3,5,20$ (clockwise from left). The nonspecific attraction favors the formation of tiling vertices, which for the regular rhombus leads to the emergence of the nonparallel ordered tiling. However, as $A$ departs enough from $\sqrt{3}$, the nonparallel ordered tiling cannot exist. Open structures result.Reuse & Permissions